D. Kotschick: Riemannian Geometry
-
Time and place: Tue and Wed 10-12 in HS A 027
-
Recitation classes: Mon 14-16 in HS B 047
-
Lecture notes
-
Contents: This is a first course in Riemannian geometry, covering the following topics: connections on vector bundles, the Euler class for rank 2 bundles and the Gauss-Bonnet theorem for abstract surfaces, pseudo-Riemannian manifolds and their Levi-Civita connections in arbitrary dimensions, the curvature tensor, geodesics, completeness, exponential map, Jacobi fields, isometries, spaces of constant curvature and other model spaces, relations between curvature and topology, for example the classical theorems of Bonnet-Myers and of Cartan-Hadamard. If time permits, further topics related to Ricci curvature and/or volume growth will be covered.
-
Intended audience: All students of mathematics and/or physics who are interested in geometry and have a working knowledge of the basic facts about smooth manifolds.
-
Prerequisites: We shall assume only a basic knowledge of smooth manifolds. It is not necessary to have attended Differentiable Manifolds last semester. That course covered more preliminary material than is needed to understand this course.
-
Main references for the course:
M. P. do Carmo: Riemannian Geometry, Birkhäuser Verlag 1992.
P. Pedersen: Riemannian Geometry, Springer Verlag 1998.
Other recommended reading:
S. Gallot, D. Hulin and J. Lafontaine: Riemannian Geometry, Springer Verlag 1987, 1990.
R. L. Bishop and R. J. Crittenden: Geometry of Manifolds, 1964, reprinted 2001 by AMS Chelsea Publishing.
I. Chavel: Riemannian Geometry: A modern introduction, Cambridge University Press 1993.
For the prerequisites:
L. Conlon: Differentiable Manifolds --- A first course, Birkhäuser Verlag 1993.
F. Warner: Foundations of Differentiable Manifolds and Lie Groups,
Springer Verlag 1983.
These references cover much more than we need.
-
Examination: There will be a written exam in July. Details about the registration for the exam will be given during the lectures.